The surfaces of many cells are viscous fluids; consequently, most membrane proteins are able to diffuse laterally, in a more or less random fashion, with diffusion coefficients typically of order 10 -~Β° cm2/sec. If a molecule (ligand) in solution outside the cell and a protein molecule on the surface (receptor) each have two or more sites at which they can interact with one another, large, branched receptor-ligand networks can form on the cell surface by virtue of the chemical interactions that surface fluidity permits. Evidence from a variety of systems indicates that such receptor clustering plays a role in the sequence of events leading to cellular activity. This paper describes a number of mathematical problems that arise in the analysis of experiments in which clustering occurs.
I begin by reviewing methods for finding the time evolution of the cluster size distribution function in terms of reaction rate constants. The methods solve an essentially infinite system of coupled nonlinear differential equations. Next, the rate constants are analyzed, the Brownian motion problems that arise in attempting to understand ligand recognition are described and relevant experimental systems are discussed. Finally the notion of ligand as a signal amplifier is introduced--an idea that emerges naturally from the requirement that receptors be clustered for a finite amount of time before a signal can be transmitted.